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My question has to do with distance spheres in $\mathbb CP^{n+1}$. I am interested in knowing what is the radius $r$ of a distance sphere $S(r)$ around a point that makes it a minimal submanifold of $\mathbb CP^{n+1}$. It is known that, in this type of geometric situation, the distance spheres have constant mean curvature and some distance sphere will be minimal, and my problem is determining at what exact radius $r_0$ that happens. I thought of two possible approaches to this problem, and they seem to provide different answers; the first gives $r_0=\arctan\sqrt{2n+1}$ and the second $r_0=\pi/4$. Although I tend to believe the second is correct, I would like to understand what is going wrong here, or what I am missing...

Distance spheres in $\mathbb CP^{n+1}$:

Consider $\mathbb CP^{n+1}$ with the standard Fubini-Study metric $g_{FS}$ and distance spheres around a point $p\in\mathbb CP^{n+1}$. These are isometric embeddings $f_r\colon S^{2n+1}\to\mathbb CP^{n+1}$, parameterized by their radius $r\in ]0,\pi/2[$, that foliate $\mathbb CP^{n+1}$, and as $r\to 0$ collapse to the (round) point $p$ and as $t\to\pi/2$, these spheres collapse to $\mathbb CP^n\subset\mathbb CP^{n+1}$. They also happen to be (principal) orbits of a cohomogeneity one action of $\mathrm{SU}(n+1)$, whose singular orbits are the fixed point $p$ and $\mathbb CP^n$.

It is well-known that the metric in such distance spheres is a Berger metric, i.e., obtained by shrinking the round metric in the direction of the Hopf fibers, see e.g. Petersen's book. In fact, one can compute the metric on the distance sphere of radius $r$ to be $$g_r:=f_r^*(g_{FS})=\sin^2 r (g+(\cos^2 r) h),$$ where we decompose the round metric as $g_{S^{2n+1}}=g+h$, such that $h$ is the component in the direction of the Hopf fiber and $g$ is the component in the directions orthogonal to the Hopf fiber.


First approach:

To find a minimal $(S^{2n+1},g_r)$ inside $\mathbb CP^{n+1}$, we compute its mean curvature, by computing its second fundamental form. According to multiple sources (e.g. Maeda's paper, p. 38 or Karcher's survey p. 220), the second fundamental form $A_r$ of $(S^{2n+1},g_r)$ is given by: $$A_r(\xi)=(2\cot 2r)\xi, \quad A_r(u)=(\cot r)u,$$ where $\xi$ is the vector tangent to the Hopf fiber and $u$ is any tangent vector orthogonal to $\xi$. Therefore, since we can form an orthonormal basis of eigenvectors of $A_r$ with $1$ vector in the direction of $\xi$ and $2n$ vectors orthogonal to $\xi$, we have that the mean curvature is the sum of the corresponding eigenvalues: $$H=2\cot2r+2n\cot r=0 \Leftrightarrow r=\arctan\sqrt{2n+1}.$$

Second approach:

According to a theorem of Hsiang (Hsiang, p. 6), a $G$-orbit is minimal iff it has extremal volume among orbits of the same type. The volume of $(S^{2n+1},g_r)$ can be computed using Fubini's Theorem, as $$Vol(S^{2n+1},g_r)=Vol(\mathbb CP^n) \ Vol(\mbox{Hopf fiber}),$$ and the Hopf fiber is a circle of length $2\pi\sin r\cos r$. Differentiating the above with respect to $r$, since the only factor depending on $r$ is the volume of the Hopf fiber, we find that $(S^{2n+1},g_r)$ has extremal volume (and is hence minimal) iff $r=\pi/4$.

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The second calculation is suspect, because the volume should be $O(r^{2n+1})$ when $r$ is small. –  Greg Kuperberg Aug 23 '12 at 6:00
    
@Greg: Of course... thank you for pointing that out! –  Renato G Bettiol Aug 23 '12 at 20:12

1 Answer 1

up vote 5 down vote accepted

A clearer and more correct way to do the second calculation is to look at the volume of the preimage of $S_r^{2n+1}$ in standard $S^{2n+3}$. The preimage is a certain torus $T_r$, and one has $$\mathop{Vol} T_r = 2\pi\mathop{Vol} S_r^{2n+1}.$$ At the same time, $$\mathop{Vol} T_r \propto (\cos r)(\sin r)^{2n+1},$$ because $T_r$ is the Riemannian Cartesian product of a circle of radius $\cos r$ and an $S^{2n+1}$ of radius $\sin r$. If you calculate when the derivative of this expression vanishes, you get agreement with your first calculation.

Addendum: You can also directly check that the scale of the $\mathbb{C}P^n$ quotient of $S^{2n+1}_r$, if you want to do the calculation that way, is $\sin r$, which yields a factor of $(\sin r)^{2n}$, in addition to the factor of $\sin 2r \propto (\sin r)(\cos r)$ from the Hopf fiber. So it all fits together.

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